On Sequential Vector Parameter Estimation

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On Sequential Vector Parameter EstimationSequential estimation of a vector of signal amplitudes in a linear signal model is considered. It is known that finding the optimal sequential estimator is an intractable problem with a general stopping time that is adapted to the complete observation history \cite{Ghosh87}. By properly restricting the search space to stopping times that are adapted to a specific subset of the complete observation history we derive the optimal sequential estimator under two different formulations of the problem. In the first formulation, the unconditional covariance of the estimator is used to assess its accuracy, as a common practice. In the second formulation, the conditional covariance is used for the same purpose, which is preferred over the former when there is an auxiliary statistic whose distribution does not depend on the parameters to be estimated \cite{Efron78}. Our analytical results show that the optimal stopping rule in the conditional formulation is a simple one-dimensional threshold rule for any number of parameters to be estimated. On the other hand, the optimal stopping rule in the unconditional formulation for $n$ parameters is characterized by a hypersurface in a $\frac{n^2+n}{2}$-dimensional space, which is found by numerically computing an $n$-dimensional integral. Hence, finding the optimal sequential estimator under the traditional unconditional formulation is not tractable even for a small number of parameters. However, under the conditional formulation the optimal sequential estimator has the same simple form for any number of parameters, allowing for decentralized implementations. We further consider the decentralized version of the conditional problem in a wireless sensor network where each sensor can only send a brief summary of its observations to a fusion center FC due to strict bandwidth and energy constraints.

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